Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that for all multiindices γ. Let (ℐ(ℝⁿ))′ and (ℐ_∞(ℝⁿ))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝⁿ))′ and (ℐ_∞(ℝⁿ))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley function of f belongs to L ^p(ℝⁿ) for some p ∊ (0,1], the condition f ∊ (ℐ_∞(ℝⁿ))′ is equivalent to that f ∊ (ℐ(ℝⁿ))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝⁿ) and ℐ_∞(ℝⁿ), also including their corresponding Hardy spaces.